50,575 research outputs found
Regulator constants of integral representations of finite groups
Let G be a finite group and p be a prime. We investigate isomorphism
invariants of -lattices whose extension of scalars to
is self-dual, called regulator constants. These were originally
introduced by Dokchitser--Dokchitser in the context of elliptic curves.
Regulator constants canonically yield a pairing between the space of Brauer
relations for G and the subspace of the representation ring for which regulator
constants are defined. For all G, we show that this pairing is never
identically zero. For formal reasons, this pairing will, in general, have
non-trivial kernel. But, if G has cyclic Sylow p-subgroups and we restrict to
considering permutation lattices, then we show that the pairing is
non-degenerate modulo the formal kernel. Using this we can show that, for
certain groups, including dihedral groups of order 2p for p odd, the
isomorphism class of any -lattice whose extension of scalars
to is self-dual, is determined by its regulator constants, its
extension of scalars to , and a cohomological invariant of
Yakovlev.Comment: 43 pages. Restated the main theorem (Thm 6.8) in terms of
-lattices as opposed to -lattices and added
Section 6.3 providing criteria for the theorem to apply. To appear in Math.
Proc. Cambridge Philos. So
P versus NP and geometry
I describe three geometric approaches to resolving variants of P v. NP,
present several results that illustrate the role of group actions in complexity
theory, and make a first step towards completely geometric definitions of
complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated
to MEGA 200
Exceptional zero formulae and a conjecture of Perrin-Riou
Let be an elliptic curve with split multiplicative reduction
at a prime . We prove (an analogue of) a conjecture of Perrin-Riou, relating
-adic BeilinsonKato elements to Heegner points in , and a
large part of the rank-one case of the MazurTateTeitelbaum exceptional
zero conjecture for the cyclotomic -adic -function of . More
generally, let be the weight-two newform associated with , let
be the Hida family of , and let be the
MazurKitagawa two-variable -adic -function attached to .
We prove a -adic GrossZagier formula, expressing the quadratic term of
the Taylor expansion of at as a non-zero
rational multiple of the extended height-weight of a Heegner point in
First-order limits, an analytical perspective
In this paper we present a novel approach to graph (and structural) limits
based on model theory and analysis. The role of Stone and Gelfand dualities is
displayed prominently and leads to a general theory, which we believe is
naturally emerging. This approach covers all the particular examples of
structural convergence and it put the whole in new context. As an application,
it leads to new intermediate examples of structural convergence and to a "grand
conjecture" dealing with sparse graphs. We survey the recent developments
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